DYNAMIC ANALYSIS OF CLAVEL S DELTA PARALLEL ROBOT

Transcription

1 Poceedngs of the IEEE Intenton onfeence on Robotcs & toton pe wn Septebe -9 DYNI NLYSIS O LVEL S DEL PRLLEL ROO Stc St & p-ocd D Deptent of echncs Unvesty Potehnc of chest Ron Soe tetve tx etons fo the geoetc netc nd dync nyss of Det pe obot e estbshed n ths ppe he pototype of ths npto s thee degee of feedo spt echns whch conssts of syste of pe chns Spposng tht the poston nd the tnston oton of the ptfo e nown n nvese dync pobe s soved sng the vt powes ethod ny soe ecsve tx etons nd soe gphs fo the oents nd the powes of the thee ctve copes e detened Key wods: obotcs npto ptfo tx dyncs Intodcton he pe obots e spt echnc stctes tht consst of netc cosed chns eney pe npto hve two ptfos One of the s ttched to the fx efeence fe he othe one cn hve bty otons n ts wospce hee obe egs de p s se obots connect the effecto whch s ttched to the ovng ptfo to the fxed ptfo he eeents of the obot e connected one to the othe by sphec jonts evote jonts o pstc jonts he pe nptos hve soe spec chctestcs wth espect to the se obots sch s: oe gd stcte hgh oentton cccy stbe fnctonng conto on the ts of veoctes nd cceetons stbe poston of the ctng systes nd good poston epettvty he pe obots e eqpped wth hydc o pnetc cttos hey hve obst constcton nd they cn ove bodes of consdebe sses nd densons wth hgh speeds hs s why the echnss whch podce tnston o sphec oton to ptfo e bsed on the concept of pe npto he ost nown ppcton s the fght sto wth sx degee of feedo whch s n fct the ogh- Stewt ptfo [Stewt 9; eet 997] he pe npto St [Hevé nd Spcno 99; eby nd on 999] nd the pe Det obot [ve 988; Zsobo-y ] eqpped wth thee engnes whch hve pe settng tn on the effecto n thee degee of feedo gene tnston oton sed n qc opetons of pc nd pce ngees (997) Wng nd ossen () deveoped the dect netc nyss of pototype of sphec npto ge Wst whch hs thee concent ottons hs ppe estbshes soe ecsve tx etons sed fo poston netc nd dync nyss fo //$7 IEEE thee degee of feedo Det obot In 988 R ve deveoped the pototype of ths obot t the Lsnne ede Poytechnc Insttte Invese geoetc ode he foowng eeents e the eeents of the topoogc stcte of one of the thee netc cosed chns of the npto espectvey: n engne n ctve evote jont n nteedy echns wth fo evote ns tht connect fo bs whch e pe two nd two nd fny pssve evote n connected to the ovng ptfo (fg) g he ve s Det Robot Let Ox y z ( ) be fx ctesn fe thee degees of feedo Det npto s ovng wth espect to ths efeence fe he npto hs thee egs he eeents of these egs hve nown densons nd sses One of the thee ctve eeents of the obot

2 s the fst body of the eg hs s hoogenos cn whch ottes bot the xs z wth the ng veocty nd the ng cceeton ε It hs the ength the ss nd the tenso of net he tnssson b s connected to the x y z ( ) fe nd t hs etve otton wth the nge so tht! nd ε! It hs the ss nd the tenso of net the on two dentc nd pe bs wth se ength otte bot the fe wth the nge hey hve so the se ss nd the se tenso of net he peog s cosed by n eeent whch hs the se ength nd ss wth Its tenso of net s hs eeent ottes wth the etve nge he ptfo of the obot s n eqte tnge he eton ( sn β ) gves the sde denson of ths tnge whch hs the ss Let s denote wth! (fg) the ng veocty of the ptfo wth espect to the neby body he foowng nges gve the nt poston of the npto: π π π α α π α β β β () Let s consde the otton nges of the thee cttos the petes whch gve the poston of the echns In the nvese geoetc pobe one cn consde tht the coodntes of the ss cente of the ptfo x y z gve the poston of the echns Psng the eg n the O wy one obtns the foowng pssng tces: θ θ α β θ θ β θθ whee one denoted [Stc 998]: θ θ θ cosα sn α θ α snα cosα θθ () cos β sn β sn cos β β β () cos sn sn cos j j j g x z O α z x ( ) g he netc sche of the echns If the othe egs nd of the echns e psed soe nogos etons cn be wtten he tnston condtons of the ptfo e gven by the foowng denttes y " x z y φ z " " b b c c nd by the foowng tces x 7 z φ φ z x I y φ φ φ y () " " b () β z 7 φ 7 y 7 z 7

3 " c o ths etons one obtns the foowng etons between nges: () Spposng tht the oton of the ss cente of the ptfo ong n epses s gven by the etons x y z x z y () t () t [ ] x y π sn t π cos t π () t cos β z cos t the nges e gven by the foowng geoetc condtons: b c b c $ whee one denoted: α ( sn β ) Veoctes nd cceetons (7) he otons of the coponent eeents of ech eg (fo expe the eg ) e chctesed by the foowng sew syetc tces [Stc nd p-ocd ] () (8) (9) ssocted to the bsote ng veoctes gven by the ecence etons -! () he veocty v of the jont s gven by the eton { v v- - } () v- ( ) he foowng tx etons gve the netc constnts: $ ( ) ()! ( ) he etons () gve the cob tx of the echns hs tx s n essent eeent fo the nyss of the obot wospce so the etons () epesent the connectvty condtons of the etve ng veoctes hese etons gve the ng veoctes s fncton of the tnston veocty of the ptfo Let s sse tht the obot hs vt oton detened by the ng veoctes v v v he chctestc vt veoctes expessed s fnctons of the poston of the obot e gven by the connectvty condtons of the etve veoctes of the oops - nd -: v v v v bv cv ( ) () v v v v v v Soe othe coptbty etons cn be obtned f one consdes sccessvey tht nd v he ng cceetons ε ε ε v ε of the eeents of the obot e gven by soe new connectvty condtons obtned by devng the etons () he foowng etons est: ε ε ( ) ε ε ε! ( ) ε $ () 8

4 ( ) If the othe two netc chns of the npto e psed nogos etons cn be esy obtned he foowng ecence etons gve the ng cceetons ε nd the cceetons γ of jonts ε γ ε ε ε ( ε ) ε [ ( γ ε ) ] () he etons () () epesent the nvese netc ode of the Det obot Eqtons of oton hee eectc engnes tht genete thee copes of oents nd whch hve the dectons of the xes z z z conto the oton of the egs of the npto he foce of net nd the estnt oent of the foces of net of the gd body e detened wth espect to the cente of the jont O On the othe hnd the chctestc vectos f nd evte the nfence of the cton of the weght g nd of othe exten nd nten foces pped to the se eeent of the obot g he oent Let s consde tht the oton of the ptfo s nown In these condtons one detenes fst the poston the veocty nd the cceeton of ech jont hen the foces nd the oent tht e ctng ech body e detened ny one cctes the oents of the ctve copes hee e thee ethods whch cn povde the se ests concenng these oents he fst one s sng the Newton-Ee cssc pocede the second one ppes the Lgnge eqtons nd tpes fos nd the thd one s bsed on the vt wo pncpe g he oent g he oent Kne nd Levnson (98) obtned soe vecto ecsve etons concenng the eqb of the genezed foces tht e pped to se obot In the nvese dync pobe n ths ppe one ppes the vt powes ethod n ode to estbsh soe ecsve tx etons fo the oents nd the powes of the thee ctve copes Soe gphs of these oents nd powes e so obtned 9

5 g he powe of the fst ctto s the vt veoctes ethod shows the dync eqb condton of the echns s tht the vt powe of the exten nten nd net foces whch s deveoped dng gene vt dspceent st be n ppyng the fndent eqtons of the pe obots dyncs obtned by St Stc () the foowng tx eton ests [ ( ) ( ) ( )] v v v v v v v (7) whee one denoted: ( ) [ ] ( ) f ε γ ε γ (8) g 7 he powe of the second ctto he etons (7) nd (8) epesent the nvese dync ode of the pe Det obot s ppcton et s consde obot whch hs the foowng chctestcs: g g g g z y x ny one obtns the gphs of the oents (fg) (fg) (fg) nd of the powes p (fg) p (fg7) p (fg8) gven by the ctve copes of the thee cttos g 8 he powe of the thd ctto oncsons Wthn the nvese poston nyss soe exct etons tht gve n e te the poston the veocty nd the cceeton of ech eeent of the pe obot hve been estbshed Usng the Newton-Ee cssc ethod whch tes nto ccont ech septe body of the echns n one hnded nd fve eqtons syste tht st be soved wod est ny the oents of the ctve copes cod be obtned he nytc cc nvoved n the Lgnge s eqtons nd tpes fos e too ong nd they

6 hve s of ng eos so the te fo nec ccs gows wth the nbe of the bodes of the echns he new ppoch bsed on the vt wo pncpe estbshes dect ecsve detenton of the vton n e te of the oents nd the powes of the ctve copes he tetve tx etons (7) nd (8) of the theoetc ode of dync ston cn be tnsfoed n ode fo the totc cond of the pe Det obot Refeences [] Stewt D (9) Ptfo wth Sx Degees of eedo Poceedngs of the Inst ech Engs 8 [] eet -P (997) Les obots pées Hees [] Hevé Spcno (99) St New oncept n Robotcs Poceedngs of the hd Int Woshop on dvnces n Robot Knetcs e [] eby on L (999) eoetc Synthess of Pe nptos of St-Le opoogy wth enetc goth IEEE Int onf on Robotcs nd toton Detot chgn [] ve R (988) Det: st Robot wth Pe eoety Poceedngs of the 8 th Int Sypos on Indst Robot Lsnne [] Zsobo-y P () Knetc nyss of ve s Det Robot o Sypos on echnss chnes nd echtoncs Snt-Hbet (onté) [7] ngees (997) ndents of Robotc echnc Systes heoy ethods nd goths Spnge-Veg New Yo [8] Wng ossen () Repesentton of the Sngty Loc of Spec ss of Sphec -dof Pe npto wth Revote cttos o Sypos on echnss chnes nd echtoncs Snt-Hbet (onté) [9] Stc St (998) heoetc echncs Edt Ddctc & Pedgogc chest [] Stc St () éthodes tcees en dynqe des écnses Scentfc etn Sees D echnc Engneeng Unvesty Potehnc of chest [] Stc St p-ocd D () On n Invese Dync Pobe n Robotcs Poceedngs of the th Int D Sypos Venn [] Kne R Levnson D (98) Dyncs heoy nd ppctons c w H SIU Stefn Pofesso D E-: stc@ctecpbo Sp Independente chest Ron RP-IORDI Dne t Rede D E-: cp@ctecpbo